Solve (x+y-z+1)(x-y+z+1)

$$(x+y-z+1)(x-y+z+1)$$

Expand

$$x^{2}+2x-y^{2}+2yz-z^{2}+1$$

Solution Steps

Apply the distributive property by multiplying each term of x+y-z+1 by each term of x-y+z+1.

$$x^{2}-xy+xz+x+yx-y^{2}+yz+y-zx+zy-z^{2}-z+x-y+z+1$$

Combine -xy and yx to get 0.

$$x^{2}+xz+x-y^{2}+yz+y-zx+zy-z^{2}-z+x-y+z+1$$

Combine xz and -zx to get 0.

$$x^{2}+x-y^{2}+yz+y+zy-z^{2}-z+x-y+z+1$$

Combine yz and zy to get 2yz.

$$x^{2}+x-y^{2}+2yz+y-z^{2}-z+x-y+z+1$$

Combine x and x to get 2x.

$$x^{2}+2x-y^{2}+2yz+y-z^{2}-z-y+z+1$$

Combine y and -y to get 0.

$$x^{2}+2x-y^{2}+2yz-z^{2}-z+z+1$$

Combine -z and z to get 0.

$$x^{2}+2x-y^{2}+2yz-z^{2}+1$$

Evaluate

$$\left(x+y-z+1\right)\left(x-y+z+1\right)$$

Solution Steps

Apply the distributive property by multiplying each term of x+y-z+1 by each term of x-y+z+1.

$$x^{2}-xy+xz+x+yx-y^{2}+yz+y-zx+zy-z^{2}-z+x-y+z+1$$

Combine -xy and yx to get 0.

$$x^{2}+xz+x-y^{2}+yz+y-zx+zy-z^{2}-z+x-y+z+1$$

Combine xz and -zx to get 0.

$$x^{2}+x-y^{2}+yz+y+zy-z^{2}-z+x-y+z+1$$

Combine yz and zy to get 2yz.

$$x^{2}+x-y^{2}+2yz+y-z^{2}-z+x-y+z+1$$

Combine x and x to get 2x.

$$x^{2}+2x-y^{2}+2yz+y-z^{2}-z-y+z+1$$

Combine y and -y to get 0.

$$x^{2}+2x-y^{2}+2yz-z^{2}-z+z+1$$

Combine -z and z to get 0.

$$x^{2}+2x-y^{2}+2yz-z^{2}+1$$